Is the set $\left\{ u\in L^{2}\left(\Omega\right):0\leq u\left(x\right)\leq1\,a.e.\right\} $ closed?

Question

If $\Omega\subset\mathbb{R}^{n}$ is bounded, closed and convex with smooth boundary, is the set $\left\{ u\in L^{2}\left(\Omega\right):0\leq u\left(x\right)\leq1\,a.e.\right\} $ closed in $L^{2}\left(\Omega\right)$?